论文标题
PICARD RANS跳跃K3表面有差的降低
Picard rank jumps for K3 surfaces with bad reduction
论文作者
论文摘要
令$ x $为一个数字字段的K3表面。我们证明，$ x $具有无限的专业，其PICARD RACK会跳跃，因此将我们以前与Shankar-Shankar-Shankar（tang）一起扩展到了$ x $可能会降低的情况。我们证明，对于一般普通的非异端家族，K3表面的曲线上的curves {\ mathbb {f}} _ p $都扩展了Maulik-Shankar-tang的先前工作。结果，我们为正交和统一的Shimura品种提供了普通的Hecke轨道猜想的新证明。
Let $X$ be a K3 surface over a number field. We prove that $X$ has infinitely many specializations where its Picard rank jumps, hence extending our previous work with Shankar--Shankar--Tang to the case where $X$ might have potentially bad reduction. We prove a similar result for generically ordinary non-isotrivial families of K3 surfaces over curves over $\overline{\mathbb{F}}_p$ which extends previous work of Maulik--Shankar--Tang. As a consequence, we give a new proof of the ordinary Hecke orbit conjecture for orthogonal and unitary Shimura varieties.
